A finite-dimensional unital associative algebra over a field admitting a linear functional whose pairing is nondegenerate. The dual numbers with functional extracting the nilpotent coefficient are an example; semisimplicity is unnecessary.
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A **Frobenius algebra** is a type of algebra that possesses both a product and a bilinear form satisfying certain conditions, making it particularly important in representation theory, algebraic topology, and quantum field theory.